1670s, "one who denies the validity of ruling power;" see anarchy + -ist. The word got a boost during the French Revolution; in 19c. it was used both of "one who advocates absence of government as a political ideal" (philosophical or scientific anarchism) and "one who seeks to overthrow violently all forms and institutions of society and government with no intention of establishing others."
Middle English -ik, -ick, word-forming element making adjectives, "having to do with, having the nature of, being, made of, caused by, similar to," from French -ique and directly from Latin -icus or from cognate Greek -ikos "in the manner of; pertaining to." From PIE adjective suffix *-(i)ko, which also yielded Slavic -isku, adjectival suffix indicating origin, the source of the -sky (Russian -skii) in many surnames. In chemistry, indicating a higher valence than names in -ous (first in benzoic, 1791).
In Middle English and after often spelled -ick, -ike, -ique. Variant forms in -ick (critick, ethick) were common in early Modern English and survived in English dictionaries into early 19c. This spelling was supported by Johnson but opposed by Webster, who prevailed.
<a href="https://www.etymonline.com/word/anarchistic">Etymology of anarchistic by etymonline</a>
Harper, D. (n.d.). Etymology of anarchistic. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/anarchistic